Extreme values of the mass distribution associated with $d$-quasi-copulas via linear programming

TL;DR

This work addresses the problem of extremal $Q$-volumes $V_Q({\mathcal B})$ for $d$-quasi-copulas over $d$-boxes ${\mathcal B}\subseteq[0,1]^d$. It introduces a grid-based relaxation that reduces the problem to linear programs via two key results (Theorems $main$ and $main$-$v2$) that guarantee extendability from finite grids to full quasi-copulas under Boundary, Monotonicity, and Lipschitz constraints. The authors solve separate linear programs for minimal and maximal volumes, obtaining results up to $d=18$ for the minimum and $d=17$ for the maximum, and show that extremal configurations occur with boxes $[a,b]^d$ (minimal) and $[a,1]^d$ (maximal), with $q_{\mathbb I}$ depending only on $||{\mathbb I}||_1$. The findings refute the conjectured minimal-volume formula $-\frac{(d-1)^2}{2d-1}$, suggest an exponential growth pattern $c2^{d}+d$, and provide explicit symmetric realizations for small dimensions, offering a roadmap for analytic proofs and extensions to broader quasi-copula classes.

Abstract

The recent survey published in Fuzzy Sets and Systems nicknamed ``Hitchhiker's Guide'' has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. Some of the open problems listed there were solved, and some conjectured one way or the other. This paper concentrates on the Open Problem 5 of this list concerning bounds on the volume of a $d$--variate quasi-copula. We disprove a recent conjecture published in the same journal on the lower bound of this volume. We also give evidence that the problem is much more difficult than suspected and provide hints about its final solution.

Figures (3)

• Figure 1: An illustration of the notation we use to denote the vertices of the box ${\mathcal{B}}$ and the values of the $d$-quasi-copula $Q$ in those vertices. For $d=2$ the box ${\mathcal{B}} = [a_1, b_1]\times [a_2, b_2]$ has four vertices $x_{0,0} = (a_1, a_2)$, $x_{1,0} = (b_1, a_2)$, $x_{0,1} = (a_1, b_2)$, $x_{1,1} = (b_1, b_2)$ that are indexed by four multi-indices $\mathbb{I}^{(1)} = (0,0)$, $\mathbb{I}^{(2)} = (1,0)$, $\mathbb{I}^{(3)} =(0,1)$ and $\mathbb{I}^{(4)} = (1, 1)$. The value of $Q$ at the vertex $x_{\mathbb{I}^{(k)}}$ is denoted by $q_{\mathbb{I}^{(k)}}$. For $d=2$ we have $q_{0,0} = Q(a_1, a_2)$, $q_{1,0} = Q(b_1, a_2)$, $q_{0,1} = Q(a_1, b_2)$, $q_{1,1} = Q(b_1, b_2)$, while for $d=3$ we have $q_{0,0,0} = Q(a_1, a_2, a_3)$, $q_{1,0, 0} = Q(b_1, a_2, a_3)$, $q_{0,1,0} = Q(a_1, b_2, a_3)$, $q_{0, 0, 1} = Q(a_1, a_2, b_3)$, $q_{1,1,0} = Q(b_1, b_2, a_3)$, $q_{1,0, 1} = Q(b_1, a_2, b_3)$, $q_{1,1,0} = Q(b_1, b_2, a_3)$, $q_{1, 1, 1} = Q(b_1, b_2, b_3)$.
• Figure 2: To construct a quasi-copula $Q$ on $[0,1]^3$ given the values $q_\mathbb{I}$ at $x_\mathbb{I}\in \mathcal{D}:=\prod_{i=1}^3\{a_i,1\}$, we first define it to be 0 on all $2$-dimensional faces $\mathcal{L}_1$, $\mathcal{L}_2,$$\mathcal{L}_3$ and prove that the extension indeed meets the requirements of a quasi-copula. In the case $\mathcal{D}:=\prod_{i=1}^3 \{a_i,b_i,1\}$ the same applies, only that the initial grid consists of $3^3=27$ points, having values $q_{\mathbb{I}_1,\mathbb{I}_2,\mathbb{I}_3}$, $\mathbb{I}_j\in \{0,1,2\}$.
• Figure 3: Blue (and orange) points represent the volumes of boxes with maximal (and minimal) volume over all $d$-quasi-copulas and all $d$-boxes. The $x$-axis represents the dimension $d$, while the $y$-axis represents the value of the volume in thee logarithmic scale with base 2.

Theorems & Definitions (25)

• Theorem 2.1
• Example 1
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Proposition 2.6